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ON THE $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -COMPLETENESS AND $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -COMPLETENESS OF MULTIFRACTAL DECOMPOSITION SETS

Research paper by L. Olsen

Indexed on: 07 Mar '18Published on: 01 Jan '18Published in: Mathematika



Abstract

The purpose of this paper twofold. Firstly, we establish $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if $\unicode[STIX]{x1D707}$ is a self-similar measure satisfying the strong separation condition and $\unicode[STIX]{x1D707}$ is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of $\unicode[STIX]{x1D707}$ defined by $$\begin{eqnarray}\bigg\{x\in \mathbb{R}^{d}\,\bigg; \,\lim _{r{\searrow}0}\frac{\log \unicode[STIX]{x1D707}(B(x,r))}{\log r}=\unicode[STIX]{x1D6FC}\bigg\}\end{eqnarray}$$ are $\unicode[STIX]{x1D6F1}_{3}^{0}$ -complete provided they are non-empty.